By Linfan Mao

A combinatorial map is a attached topological graph cellularly embedded in a floor. This monograph concentrates at the automorphism team of a map, that is concerning the automorphism teams of a Klein floor and a Smarandache manifold, additionally utilized to the enumeration of unrooted maps on orientable and non-orientable surfaces. a couple of effects for the automorphism teams of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are chanced on. An straight forward class for the closed s-manifolds is located. Open difficulties with regards to the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries also are awarded during this monograph for the extra functions of the combinatorial maps to the classical arithmetic.

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**Extra resources for Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Partially Post-Doctoral Research for the Chinese Academy of Sciences)**

**Example text**

Whence, not loss of generality, we only need to consider the voltage xij on the common boundary among the face fi and fj for 1 ≤ i, j ≤ n. Then the voltage assignment on the n faces are ϑ(f1 ) = x12 x13 · · · x1n , ϑ(f2 ) = x21 x23 · · · x2n , ·················· ϑ(fn ) = xn1 xn2 · · · xn(n−1) . We wish to find an assignment on M which can enables us to get as many faces as possible with the voltage of order p1 . Not loss of generality, we can choose −1 ϑp1 (f1 ) = 1ZN in the first. To make ϑp1 (f2 ) = 1ZN , choose x23 = x−1 13 , · · · , x2n = x1n .

Proof Assume that f1 , f2 , · · · , fn , where,n = φ(M), are the n faces of the map M. By the definition of voltage assignment, if x, βx or x, αβx appear on one face fi , 1 ≤ i ≤ n altogether, then they contribute to ϑ(fi ) only with ϑ(x)ϑ−1 (x) = 1ZN . Whence, not loss of generality, we only need to consider the voltage xij on the common boundary among the face fi and fj for 1 ≤ i, j ≤ n. Then the voltage assignment on the n faces are ϑ(f1 ) = x12 x13 · · · x1n , ϑ(f2 ) = x21 x23 · · · x2n , ·················· ϑ(fn ) = xn1 xn2 · · · xn(n−1) .

Therefore, for ∀x ∈ Xα,β , we have Θg(x) = (g(x), Pg(x)) and gΘ(x) = g(x, Px) = (g(x), Pg(x)). Whence, we get that for ∀x ∈ Xα,β , Θg(x) = gΘ(x). , gΘg −1 = Θ. that Since for ∀x ∈ Xα,β , gΘg −1(x) = (g(x), Pg(x)) and Θ(x) = (x, P(x)), we have (g(x), Pg(x)) = (x, P(x)). That is, g is a conformal mapping. 2 The non-Euclid area on a map For a given voltage map (M, G), its non-Euclid area µ(M, G) is defined by Chapter 2 43 On the Automorphisms of a Klein Surface and a s-Manifold µ(M, G) = 2π(−χ(M) + (−1 + m∈O(F (M )) 1 )).

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